Logarithm Calculator

Calculate log₁₀, ln, log₂ and custom-base logarithms with antilog

What is it and how does it work?

A logarithm answers the question "to what power must the base be raised to get this number?" — log₂(8) = 3 because 2³ = 8. Three bases dominate in practice: base-10 (log, used in science and the decibel scale), base-e (ln, the natural logarithm, used in calculus and growth models), and base-2 (log₂, used in information theory and computing). This calculator handles all three plus any arbitrary base you specify.

Logarithms appear constantly in algorithm analysis (O(log n) complexity), audio engineering (decibels), signal processing, statistics (log-normal distributions) and financial mathematics (compound growth). The change-of-base formula — log_b(x) = ln(x) / ln(b) — means any base reduces to natural log, which is how this tool computes arbitrary bases.

Common use cases

• Algorithm analysis: how many times must you halve 1,000,000 before reaching 1? log₂(1,000,000) ≈ 19.9, so about 20 — the depth of a binary search tree.
• Decibels: a sound at 10,000 times the reference intensity is 10 × log₁₀(10000) = 40 dB.
• Compound growth: an investment that doubles every 7 years — what annual rate? ln(2)/7 ≈ 9.9% per year (continuous compounding).

Frequently asked questions

What is the difference between log and ln?

log (or log₁₀) is base-10 logarithm, used in engineering and the decibel scale. ln is the natural logarithm, base-e (≈ 2.71828), the one calculus treats as "natural" because its derivative is 1/x. In pure maths, "log" often means ln; in engineering it usually means log₁₀ — always check the context.

Why can't I take the log of a negative number or zero?

Logarithms are only defined for positive real numbers. No real power of any positive base produces a negative result or zero. The complex-number extension exists but is multi-valued and rarely needed in practice.

What is log(1) regardless of base?

Always 0. Any base raised to the power 0 equals 1, so the logarithm of 1 is 0 for every base.

How do I convert between bases?

Use the change-of-base formula: log_b(x) = log_c(x) / log_c(b) for any convenient base c. For example, log₂(100) = log₁₀(100) / log₁₀(2) = 2 / 0.301 ≈ 6.644.

Math

Percentage Calculator · Age Calculator · Date Difference · Scientific Calculator · BMI Calculator · Fibonacci Generator